$k$-free lattice points in random walks

Let $\mathbb{Z}^2$ be the two-dimensional integer lattice. For an integer $k\geq 1$, a non-zero lattice point is $k$-free if the greatest common divisor of its coordinates is a $k$-free number. We consider the proportions of $k$-free and twin $k$-free lattice points on a path of an $\alpha$-random walker in $\mathbb{Z}^2$. Using the second-moment method and tools from analytic number theory, we prove that these two proportions are $1/\zeta(2k)$ and $\prod_{p}(1-2p^{-2k})$, respectively, where $\zeta$ is the Riemann zeta function and the infinite product takes over all primes.